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A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. (English) Zbl 0818.30004
It is well known that the conformal mapping of a simply [or doubly] connected domain $$D$$ onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the ‘charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by $$G(z)= \sum^ N_{i= 1} Q_ i\log | z- \zeta_ i|$$ with unknown coefficients $$Q_ i$$, where the charge points $$\zeta_ i$$ lie outside $$\overline D$$. To determine the $$Q_ i$$ it is required that $$G(z_ j)= b(z_ j)$$ for $$N$$ collocation points $$z_ j\in \partial D$$. The error is defined by $$E_ G= \max| G(z_{j+{1\over 2}})- b(z_{j+{1\over 2}})|$$, where $$z_{j+{1\over 2}}\in \partial D$$ are intermediate points. The determination of $$G$$ thus leads to the solution of a $$N\times N$$ linear system for the $$Q_ i$$. – The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the $$N\times N$$ matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on $$\partial D$$ occur.
Reviewer: D.Gaier (Gießen)

##### MSC:
 30C30 Schwarz-Christoffel-type mappings
##### Keywords:
charge simulation method
Full Text:
##### References:
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